## January 31, 2013

## Suprema of certain Gaussian processes

## Adam Harper, Cambridge University and CRM

I will describe a number-theoretic problem that can be attacked using sharp quantitative estimates for the supremum of a certain Gaussian process. Then I will explain how one can obtain such estimates, using a two step "conditioning and comparison" procedure. In particular, this approach allows one to identify the expectation of the supremum up to second order terms. If time allows I will also describe applications to some other Gaussian processes.

## February 7, 2013

## Infinite Geodesics in 2D First-Passage Percolation

## Jack Hanson, Princeton University

First-passage percolation is a model for a random metric. Several longstanding problems involve the existence of infinite geodesics for this metric, as well as their asymptotic directional properties: for instance, do infinite geodesics have asymptotic direction, and is the geodesic corresponding to a given direction unique? C. Newman and collaborators made progress in this area in the 90's under strong assumptions on the passage time distribution and limiting shape. I will explain recent work with M. Damron which provides a partial resolution of these questions under general assumptions. A major part of the analysis is the development of a framework for analyzing Busemann functions, which represent distances to a point "at infinity."

## February 28, 2013

## Operator limits of random matrices

## Benedek Valko, Wisconsin

By the Hilbert-Polya conjecture the critical zeros of the Riemann zeta function correspond to the eigenvalues of a self adjoint operator. By a conjecture of Dyson and Montgomery the critical zeros (after a certain rescaling) look like the bulk eigenvalue limit point process of the Gaussian Unitary Ensemble. It is natural to ask if this point process can be described as the spectrum of a random self adjoint operator. We show that this is indeed the case: in fact for any beta>0 the bulk limit of the Gaussian beta ensemble can be obtained as the spectrum of a self adjoint random differential operator.
(Joint with Balint Virag)

## March 28, 2013

## Smoothness properties for some infinite-dimensional heat kernel measures

## Tai Melcher, University of Virginia

Smoothness is a fundamental principle in the study of measures on infinite- dimensional spaces, where an obvious obstruction to overcome is the lack of an infinite-dimensional Lebesgue or volume measure. Canonical examples of smooth measures include those induced by a Brownian motion, both its end point distribution and as a real-valued path. More generally, any Gaussian measure on a Banach space is smooth. Heat kernel measure is the law of a Brownian motion on a curved space, and as such is the natural analogue of Gaussian measure there. We will discuss some recent smoothness results for these measures on certain classes of infinite-dimensional groups, including in some degenerate settings. Some parts of this talk are joint work with Fabrice Baudoin, Daniel Dobbs, and Masha Gordina.

## April 4, 2013

## Limiting spectral distrbution of patterned matrices

## Arup Bose, ISI

We present a unified approach to establishing limiting spectral distribution(LSD) of patterned matrices via the moment method. This allows us to demonstrate relatively short proofs for the LSDs of common matrices (Wigner, Toeplitz, Hankel, Reverse Circulant, Symmetric Circulant) and provide insight into the nature of different LSDs and their interrelations. The method is applicable to matrices with appropriate dependent entries, banded matrices (including triangular matrices) and matrices of the form $A_p=\frac{1}{n}XX^\prime$ where $X$ is a $p\times n$ matrix with real entries and $p\to \infty$ with $n=n(p)\to\infty$ and $p/n \to y$ with $0\leq y <\infty$. The sample variance covariance matrix being a particular example of the latter. We can also establish the existence of the LSD of the sample autocovariance matrix and the joint convergence of several copies of different patterned matrices by this approach. It raises interesting questions about the class of patterns for which LSD exists and the nature of the possible limits. In many cases the LSDs are not known in any explicit forms and so deriving probabilistic properties of the limit are also interesting issues.

## April 11, 2013

## Product-form Invariant Measures for Brownian Motion with Drift
Satisfying a Skew-symmetry Type Condition

## Janosch Ortmann, University of Toronto

Motivated by recent developments on positive-temperature
polymer models we propose a generalisation of reflected Brownian motion
(RBM) in a polyhedral domain. Our process is obtained by replacing the
singular drift on the boundary by a continuous one which depends, via a
potential U, on the position of the process relative to the domain. We
show that our generalised process has an invariant measure in product
form, under a certain skew-symmetry condition that is independent of the
choice of potential. Applications include TASEP-like particle systems,
generalisations of Brownian motion with rank-dependent drift and
diffusions connected to the generalised Pitman transform.
The talk is based on joint work with Neil O'Connell.